report - CS229
Gaussian Process Regression with K-means Clustering for Very Short-Term Load Forecasting of Individual Buildings at Stanford
1
Carol Hsin
Abstract—The objective of this project is to return expected electricity load in kiloWatts (kW) given a building and a time interval. The raw data consists the mean kW of 15 minute intervals for two years from 122 meters corresponding to 109 buildings at Stanford resulting in a data set of 70176 observations. The data was cleaned and then split into training and test sets. K-means clustering was used to split the training set into more similar sets. Test dates and test buildings was clustered clusters and clusters were used to make the prediction on test dates from given clusters. The baseline model is an auto regressive model. GPR models with 7 days prior and GPR models based on clustering was also completed. Error analysis was done using Root Mean Squared Error.
I. I NTRODUCTION Unlike other products, electricity isn’t easily stored, so producers must anticipate and meet the maximum demand, also known as peak load, at any given time or outages will ensue. Since consuming fuel and running electrical power generators isn’t free, producers also need to reduce operational costs by keeping the generation reserve at the minimum required to meet demand. While it would be suboptimal to be producing at peak if most of the electricity generated will be wasted, it would also be problematic for outages to occur due to inadequate supply. Thus, accurate electrical forecasts are essential to energy management, production, transmission, and distribution because these forecasts allow dispatchers to make decisions on the optimal, realtime scheduling of electricity production between power plants or generation units. A. Gaussian Process Regression (GPR) A Gaussian process is a collection of random variables of which the joint distribution of any subset of the variables are Gaussian. It can be viewed as a multivariate Gaussian in which we have infinitely many variables, which basically corresponds to a function because we can think of a function as being an infinitely long vector that returns a value based on the input x. Thus, a Gaussian process can be used to describe a distribution over functions.[1] For clarity, let us explicitly state the differences between a Gaussian distribution and a Gaussian process. A Gaussian distribution is fully specified by a mean vector, µ, and a covariance matrix Σ. f = (f1 , . . . , f2 )> ∼ N (µ, Σ) for indexes i = 1, . . . , n
In contrast, a Gaussian process is fully specified by a mean function m(x) and covariance function k(x, x0 ) with x as the indexes. We can think of the mean function as an infinitely long vector and the covariance function as an infinite-by-infinite dimensional matrix. Gaussian processes with finite index sets would just be Gaussian distributions. f (x) ∼ GP(m(x), k(x, x0 )) for indexes x Gaussian distributions are nice to work with because of their special properties, particularly the marginalization property because that allows us to get finite dimensional answers to finite dimensional questions from the infinite dimensional process.[1] Recall that the marginal of a joint Gaussian is Gaussian and that formulating the marginal distribution is just taking the corresponding sub-blocks from the mean and covariance matrix of the joint distribution: xA µA ΣAA ΣAB , ) ∼ N( xB µB ΣBA ΣBB Z p(xA ) = p(xA , xB ; µ, Σ) dx xB ∈Rn
xA ∼ N (µA , ΣAA ) This property means that to make a prediction with on a test point x∗ , so one can just take the relevant sub-block of the covariance matrix of the (n + 1)dimensional joint distribution (n training points and 1 test point). Another useful key property is that the covariance matrix corresponding to a multivariate Gaussian distribution is positive semidefinite, which is also the necessary and sufficient condition for a matrix K to be a Mercer kernel. Thus, any valid kernel function can be used as a covariance function and we can apply the ‘kernel trick’ to reduce computation time. Generally, the kernel is the most important aspect of a Gaussian process and choosing the right kernel to capture the correlations in the data is where most of the work will be. Specifically, if we assume zero mean, which is a commonly used assumption, then the Gaussian process is completely defined by the covariance function. In this project, we are only interested in the case of predicting using noisy observations y = f (x) + ε where f (x) is unknown and noise is modeled as ε ∼ N (0, σn2 ). Thus the covariance of y is cov(y) = K(X, X) + σn2 I
Fig. 1: Plot of a rejected building
Fig. 2: Comparison to non-rejected data
In the data cleaning process, meters servicing the same buildings were merged and any meters serviced more than one building was removed. This process f∗ |X, y, X∗ ∼ N (f¯∗ , cov(f∗ )) reduced the data from 122 to 93 variables. All 93 remaining were plotted and analyzed to ref¯∗ = E[f∗ |X, y, X∗ ] = K(X∗ , X)[K + σn2 I]−1 y move anomalies, especially buildings with missing data cov(f∗ ) = K(X∗ , X∗ )−K(X∗ , X)[K+σn2 I]−1 K(X, X∗ ) that was replaced with a number as a stand-in. This reduced the data to 71 variables for a 70080x71 data matrix (365 days/year * 24 hours/day * 4 15-mins/60II. DATA D ESCRIPTION mins * 2 years = 70080). The raw data obtained from Stanford Facilities consists of the mean kW of 15 minute intervals for two B. Dividing the Data years from 122 meters corresponding to 109 buildings The data was divided such that only the last half at Stanford resulting in a data set of 70176 observations. would be part of the test set. The trick here was that Each building may be serviced by multiple meters and since the plan was to see how well similar buildings each meter may be servicing multiple buildings. The could help predict the load of one building, there needed data ranges from October 28, 2013 at 12:00:00 AM to to be a way to group the buildings. K-means would October 28, 2015 at 10:30:00 PM. do this, but personal computers do not have enough memory required for k-means on a 35088x71 (70176/2 = 35088). To solve this, the dates were clustered (more A. Data Exploration and Cleaning in K-Mean section). Preliminary data exploration revealed there are 8 For each experiment, new models were trained per missing data points within the period being examined. test date per test building and only data from past wrt the This was discovered while formatting the time. After test date/time was used to obtain predictions. Basically, looking at the data file, missing points occurred on for one building chosen from each of the building cluster March 9, 2014 and March 8, 2015 between 01:45:00 and one day from each of the date clusters, we got a total PST and 03:00:00 PDT, so for about one hour starting of 4 test buildings and 8 test dates, so 32 models per at 2AM. This corresponds to daylight savings time, so experiment. There are 3 experiments (see Experiments the data for that period is missing because the hours section). don’t actually exist and this was confirmed because Nov and after defining K = K(X, X) to simplify notation, the key predictive equations for GPR is:[1]
3, 2013 and Nov 2, 2014 has extra data points. The missing points around March were extrapolated using the mean of the points before and after while the extra points were merged by averaging every consecutive two points.
III. K- MEANS C LUSTERING K-means clustering was performed to get similar dates with which to cluster the buildings. R’s kmeans and plotting functions were used in this section. 2
A. Clustering Dates To cluster dates, the 35040x71 matrix was reduced to a 35040x1 mean vector (mean across the 71 buildings). Then the vector was reshaped so that each row was a date (1:365) and the columns were the time intervals (1:96 for 24 hours * 4 15-min/60-min).
1 1 . .. . → [365x96] → [8x96] → .. → [71x768] 768 35040
The resulting 71x768 matrix was clustered by rows and the following cluster dendrogram was used to determine the appropriate K-value. See figure on page 4 for building cluster dendrogram. A k-value of 4 was chosen by the usual distance metric because it was the highest number of clusters that was reasonable given the data.
IV.
E XPERIMENTS AND M ODELS
A. Baseline Model: Autoregressive Linear Model For the 32 baseline models, the data point is predicted based only on data from the past. An AR model was completed for each building and for each test date based on 30 days prior to the test date. R’s auto.arima was used to process each data vector. An input vector of 30*96=2880 points prior to a test date was used to predict the output vector of 96 for the next 96 time points in the test date. This was done using a for loop, so only one time point (of the 96) was predicted per run and the prediction saved into a vector, then the next test point would be predicted with the same 30 days but also with the previous time point from the data.
The resulting matrix was fed into R’s kmeans function to cluster the rows using the euclidean distance (each point seen as a 96-dimensional vector). The resulting hierarchical cluster dendrogram revealed 7 possible Kvalues based on euclidean distances calculated. While each point is in 96 dimensional space, a visualization using usual discriminant coordinates was created for better analysis. For some K-values, it was clear which days were being clustered, e.g. winter vacation in K=4 was cluster 4. A k value of 8 was picked because the distances were reasonable and the goal was to get as many clusters as the data would allow.
B. Clustering Buildings The dates clusters were used to cluster the buildings. The clusters were used to turn the 35040x71 data matrix into a 71x768 matrix by splitting dates of the building vector (35040x1) into the cluster groups and then taking the mean of each group and then reshapping that into a vector to be inserted into the k-means clustering building matrix. 3
date 001 243 107 337 341 298 076 259
AR Building 27 trainRMSE testRMSE 3.11562406136734 5.92908086302396 2.80726350414573 27.1727170432979 3.57288754308325 25.0050035248015 3.60722058939789 19.6730977413011 3.64815270136514 6.34749946969175 3.17871587441246 13.7552846364742 3.00282373966765 16.4055959407633 2.86685920192124 22.5292421013259
date 001 243 107 337 341 298 076 259
AR Building 41 trainRMSE testRMSE 11.3137171108784 1.63099169210129 1.74704263141441 16.5290857191283 15.8048459682757 68.6245047572578 16.1212470934128 71.9370965560381 16.0362318459376 53.1374392548315 16.6870934592130 40.6767609910458 16.1701762503520 37.6412048858712 5.48500264308082 50.8484470522624
date 001 243 107 337 341 298 076 259
AR Building 50 trainRMSE testRMSE 3.38363367869690 11.8094774861282 3.73769993365999 17.0612639134373 4.57362220540115 37.5793096579895 3.94042377491766 48.1890094398847 3.88495198175363 10.8304380342995 4.21250373061003 14.1253591604769 3.97807505678671 17.8563097039936 3.56124275581864 36.8535230182934
date 001 243 107 337 341 298 076 259
AR Building 53 trainRMSE testRMSE 15.9652386786304 25.3668404088917 10.8554679708035 54.8878136779890 16.6909585241200 61.2952412778726 13.8077246997190 59.4570004960659 15.3236243694047 51.9888134588392 12.7511734577707 20.4150748457536 11.3992393682145 36.7204665273363 11.7719471437517 107.842217763288
B. GPR with 7 days prior The 32 models used an input vector of size 7*96 = 672 to predict the output vector of size 96. Like in AR, a for loop was used and a vector was used to keep track of each time point predicted. For this and all models using GPR, the covariance matrix used was Squared Error and the hyperparameters were tuned in part by using the GPML Matlab toolkit. [2] 4
date 001 243 107 337 341 298 076 259
Building27errs.csv trainRMSE testRMSE 1.8478 3.2777 2.0239 4.1934 2.5364 4.1437 3.0882 9.7364 2.9625 4.8741 1.8401 3.1328 1.2381 6.0168 2.1532 3.5589
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 2.1173 2.1642 2.1173 2.8415 15.4112 25.6011 10.4088 25.5864 15.2884 18.6595 14.2461 20.6533 7.9567 19.6042 7.4125 22.0174
date 001 243 107 337 341 298 076 259
Building50errs.csv trainRMSE testRMSE 1.9757 4.0314 2.7139 6.2222 2.7835 5.4576 2.9771 6.8797 3.1093 4.7846 3.6101 4.5342 1.6541 5.5567 3.0225 7.3988
date 001 243 107 337 341 298 076 259
Building53errs.csv trainRMSE testRMSE 13.0682 26.4726 6.0205 14.1705 6.4943 12.7095 11.4501 26.5387 15.8296 16.2554 6.4616 10.8375 3.5744 10.2883 6.4255 12.1282
date 001 243 107 337 341 298 076 259
V. A NALYSIS The models using the clustering did not perform as well as expected compared to the models using just the historical data of that particular building being analyzed. This may be that a building’s profile is quite unique and surround buildings may not be good features for predictions. It might also be the case that the clustering could be better. Further plots and analysis of the clustered dates were completed. The data to suggests that the days being clustered do follow a logic in that each has a pattern. The plots of cluster k=8 with cluster 1 is displayed. The mean Kw corresponds to the mean vector of the matrix from the first half of the data, as described in the k-means section.
C. GPR with similar building The 32 models used an input matrix of size (7*96 = 672)x(number of buildings in cluster) to predict the output vector of size 96. Each of the test buildings were from a building cluster, so their respective building clusters were used as the input. Thus, a vector of the time period (672) from each building in the test building’s cluster that is not the test building was used and the input matrices’ sizes differed depending on the number of buildings in the cluster. date 001 243 107 337 341 298 076 259
Building27errs.csv trainRMSE testRMSE 2.2500 2.8123 0.6901 7.9039 0.8641 6.9301 1.0063 17.3021 0.7230 6.7764 2.1044 4.9508 0.1623 6.5125 0.0086 6.5019
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 0.2706 1.5223 0.0758 3.2870 0.0972 17.0242 0.0653 24.6378 0.0016 16.3858 0.0005 19.8603 0.0041 20.5427 0.0018 19.8467
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 2.3962 4.9584 3.8441 7.2553 5.2072 9.3673 3.9320 10.9341 4.6400 6.8721 3.6479 7.3121 0.8671 9.5621 3.8320 11.0443
Building50errs.csv trainRMSE testRMSE 16.2173 17.9709 7.3066 19.9443 9.6251 14.6105 13.0713 21.4529 9.3861 19.3867 9.2154 9.7162 0.0250 16.5415 8.7231 13.4898
Errors might also be due to having only 7 days. The author wanted to use more than 7 days, but GPR is a non-parameteric model, which means it is very memory intensive and Matlab gave errors that ‘the array exceeds maximum array size preference’. More data points also slows the for loop from 1:96 that retrains a model per run to be slow and given 32 models to per experiment. VI. C ONCLUSION While the results from the experiments seem to suggest that similar days are not quite that important, it might be due to the clustering of the buildings, so more work needs to be done to understand why similar buildings do not have a high predictive value.
[1] [2]
5
R EFERENCES Carl Edward Rasmussen and Chris Williams. Gaussian Processes for Machine Learning. Cambridge, Massachusetts: The MIT Press, 2006. Carl Edward Rasmussen and Chris Williams. The GPML Toolbox version 3.5. manual for GPML Matlab Code version 3.6. July 2015.
1
Carol Hsin
Abstract—The objective of this project is to return expected electricity load in kiloWatts (kW) given a building and a time interval. The raw data consists the mean kW of 15 minute intervals for two years from 122 meters corresponding to 109 buildings at Stanford resulting in a data set of 70176 observations. The data was cleaned and then split into training and test sets. K-means clustering was used to split the training set into more similar sets. Test dates and test buildings was clustered clusters and clusters were used to make the prediction on test dates from given clusters. The baseline model is an auto regressive model. GPR models with 7 days prior and GPR models based on clustering was also completed. Error analysis was done using Root Mean Squared Error.
I. I NTRODUCTION Unlike other products, electricity isn’t easily stored, so producers must anticipate and meet the maximum demand, also known as peak load, at any given time or outages will ensue. Since consuming fuel and running electrical power generators isn’t free, producers also need to reduce operational costs by keeping the generation reserve at the minimum required to meet demand. While it would be suboptimal to be producing at peak if most of the electricity generated will be wasted, it would also be problematic for outages to occur due to inadequate supply. Thus, accurate electrical forecasts are essential to energy management, production, transmission, and distribution because these forecasts allow dispatchers to make decisions on the optimal, realtime scheduling of electricity production between power plants or generation units. A. Gaussian Process Regression (GPR) A Gaussian process is a collection of random variables of which the joint distribution of any subset of the variables are Gaussian. It can be viewed as a multivariate Gaussian in which we have infinitely many variables, which basically corresponds to a function because we can think of a function as being an infinitely long vector that returns a value based on the input x. Thus, a Gaussian process can be used to describe a distribution over functions.[1] For clarity, let us explicitly state the differences between a Gaussian distribution and a Gaussian process. A Gaussian distribution is fully specified by a mean vector, µ, and a covariance matrix Σ. f = (f1 , . . . , f2 )> ∼ N (µ, Σ) for indexes i = 1, . . . , n
In contrast, a Gaussian process is fully specified by a mean function m(x) and covariance function k(x, x0 ) with x as the indexes. We can think of the mean function as an infinitely long vector and the covariance function as an infinite-by-infinite dimensional matrix. Gaussian processes with finite index sets would just be Gaussian distributions. f (x) ∼ GP(m(x), k(x, x0 )) for indexes x Gaussian distributions are nice to work with because of their special properties, particularly the marginalization property because that allows us to get finite dimensional answers to finite dimensional questions from the infinite dimensional process.[1] Recall that the marginal of a joint Gaussian is Gaussian and that formulating the marginal distribution is just taking the corresponding sub-blocks from the mean and covariance matrix of the joint distribution: xA µA ΣAA ΣAB , ) ∼ N( xB µB ΣBA ΣBB Z p(xA ) = p(xA , xB ; µ, Σ) dx xB ∈Rn
xA ∼ N (µA , ΣAA ) This property means that to make a prediction with on a test point x∗ , so one can just take the relevant sub-block of the covariance matrix of the (n + 1)dimensional joint distribution (n training points and 1 test point). Another useful key property is that the covariance matrix corresponding to a multivariate Gaussian distribution is positive semidefinite, which is also the necessary and sufficient condition for a matrix K to be a Mercer kernel. Thus, any valid kernel function can be used as a covariance function and we can apply the ‘kernel trick’ to reduce computation time. Generally, the kernel is the most important aspect of a Gaussian process and choosing the right kernel to capture the correlations in the data is where most of the work will be. Specifically, if we assume zero mean, which is a commonly used assumption, then the Gaussian process is completely defined by the covariance function. In this project, we are only interested in the case of predicting using noisy observations y = f (x) + ε where f (x) is unknown and noise is modeled as ε ∼ N (0, σn2 ). Thus the covariance of y is cov(y) = K(X, X) + σn2 I
Fig. 1: Plot of a rejected building
Fig. 2: Comparison to non-rejected data
In the data cleaning process, meters servicing the same buildings were merged and any meters serviced more than one building was removed. This process f∗ |X, y, X∗ ∼ N (f¯∗ , cov(f∗ )) reduced the data from 122 to 93 variables. All 93 remaining were plotted and analyzed to ref¯∗ = E[f∗ |X, y, X∗ ] = K(X∗ , X)[K + σn2 I]−1 y move anomalies, especially buildings with missing data cov(f∗ ) = K(X∗ , X∗ )−K(X∗ , X)[K+σn2 I]−1 K(X, X∗ ) that was replaced with a number as a stand-in. This reduced the data to 71 variables for a 70080x71 data matrix (365 days/year * 24 hours/day * 4 15-mins/60II. DATA D ESCRIPTION mins * 2 years = 70080). The raw data obtained from Stanford Facilities consists of the mean kW of 15 minute intervals for two B. Dividing the Data years from 122 meters corresponding to 109 buildings The data was divided such that only the last half at Stanford resulting in a data set of 70176 observations. would be part of the test set. The trick here was that Each building may be serviced by multiple meters and since the plan was to see how well similar buildings each meter may be servicing multiple buildings. The could help predict the load of one building, there needed data ranges from October 28, 2013 at 12:00:00 AM to to be a way to group the buildings. K-means would October 28, 2015 at 10:30:00 PM. do this, but personal computers do not have enough memory required for k-means on a 35088x71 (70176/2 = 35088). To solve this, the dates were clustered (more A. Data Exploration and Cleaning in K-Mean section). Preliminary data exploration revealed there are 8 For each experiment, new models were trained per missing data points within the period being examined. test date per test building and only data from past wrt the This was discovered while formatting the time. After test date/time was used to obtain predictions. Basically, looking at the data file, missing points occurred on for one building chosen from each of the building cluster March 9, 2014 and March 8, 2015 between 01:45:00 and one day from each of the date clusters, we got a total PST and 03:00:00 PDT, so for about one hour starting of 4 test buildings and 8 test dates, so 32 models per at 2AM. This corresponds to daylight savings time, so experiment. There are 3 experiments (see Experiments the data for that period is missing because the hours section). don’t actually exist and this was confirmed because Nov and after defining K = K(X, X) to simplify notation, the key predictive equations for GPR is:[1]
3, 2013 and Nov 2, 2014 has extra data points. The missing points around March were extrapolated using the mean of the points before and after while the extra points were merged by averaging every consecutive two points.
III. K- MEANS C LUSTERING K-means clustering was performed to get similar dates with which to cluster the buildings. R’s kmeans and plotting functions were used in this section. 2
A. Clustering Dates To cluster dates, the 35040x71 matrix was reduced to a 35040x1 mean vector (mean across the 71 buildings). Then the vector was reshaped so that each row was a date (1:365) and the columns were the time intervals (1:96 for 24 hours * 4 15-min/60-min).
1 1 . .. . → [365x96] → [8x96] → .. → [71x768] 768 35040
The resulting 71x768 matrix was clustered by rows and the following cluster dendrogram was used to determine the appropriate K-value. See figure on page 4 for building cluster dendrogram. A k-value of 4 was chosen by the usual distance metric because it was the highest number of clusters that was reasonable given the data.
IV.
E XPERIMENTS AND M ODELS
A. Baseline Model: Autoregressive Linear Model For the 32 baseline models, the data point is predicted based only on data from the past. An AR model was completed for each building and for each test date based on 30 days prior to the test date. R’s auto.arima was used to process each data vector. An input vector of 30*96=2880 points prior to a test date was used to predict the output vector of 96 for the next 96 time points in the test date. This was done using a for loop, so only one time point (of the 96) was predicted per run and the prediction saved into a vector, then the next test point would be predicted with the same 30 days but also with the previous time point from the data.
The resulting matrix was fed into R’s kmeans function to cluster the rows using the euclidean distance (each point seen as a 96-dimensional vector). The resulting hierarchical cluster dendrogram revealed 7 possible Kvalues based on euclidean distances calculated. While each point is in 96 dimensional space, a visualization using usual discriminant coordinates was created for better analysis. For some K-values, it was clear which days were being clustered, e.g. winter vacation in K=4 was cluster 4. A k value of 8 was picked because the distances were reasonable and the goal was to get as many clusters as the data would allow.
B. Clustering Buildings The dates clusters were used to cluster the buildings. The clusters were used to turn the 35040x71 data matrix into a 71x768 matrix by splitting dates of the building vector (35040x1) into the cluster groups and then taking the mean of each group and then reshapping that into a vector to be inserted into the k-means clustering building matrix. 3
date 001 243 107 337 341 298 076 259
AR Building 27 trainRMSE testRMSE 3.11562406136734 5.92908086302396 2.80726350414573 27.1727170432979 3.57288754308325 25.0050035248015 3.60722058939789 19.6730977413011 3.64815270136514 6.34749946969175 3.17871587441246 13.7552846364742 3.00282373966765 16.4055959407633 2.86685920192124 22.5292421013259
date 001 243 107 337 341 298 076 259
AR Building 41 trainRMSE testRMSE 11.3137171108784 1.63099169210129 1.74704263141441 16.5290857191283 15.8048459682757 68.6245047572578 16.1212470934128 71.9370965560381 16.0362318459376 53.1374392548315 16.6870934592130 40.6767609910458 16.1701762503520 37.6412048858712 5.48500264308082 50.8484470522624
date 001 243 107 337 341 298 076 259
AR Building 50 trainRMSE testRMSE 3.38363367869690 11.8094774861282 3.73769993365999 17.0612639134373 4.57362220540115 37.5793096579895 3.94042377491766 48.1890094398847 3.88495198175363 10.8304380342995 4.21250373061003 14.1253591604769 3.97807505678671 17.8563097039936 3.56124275581864 36.8535230182934
date 001 243 107 337 341 298 076 259
AR Building 53 trainRMSE testRMSE 15.9652386786304 25.3668404088917 10.8554679708035 54.8878136779890 16.6909585241200 61.2952412778726 13.8077246997190 59.4570004960659 15.3236243694047 51.9888134588392 12.7511734577707 20.4150748457536 11.3992393682145 36.7204665273363 11.7719471437517 107.842217763288
B. GPR with 7 days prior The 32 models used an input vector of size 7*96 = 672 to predict the output vector of size 96. Like in AR, a for loop was used and a vector was used to keep track of each time point predicted. For this and all models using GPR, the covariance matrix used was Squared Error and the hyperparameters were tuned in part by using the GPML Matlab toolkit. [2] 4
date 001 243 107 337 341 298 076 259
Building27errs.csv trainRMSE testRMSE 1.8478 3.2777 2.0239 4.1934 2.5364 4.1437 3.0882 9.7364 2.9625 4.8741 1.8401 3.1328 1.2381 6.0168 2.1532 3.5589
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 2.1173 2.1642 2.1173 2.8415 15.4112 25.6011 10.4088 25.5864 15.2884 18.6595 14.2461 20.6533 7.9567 19.6042 7.4125 22.0174
date 001 243 107 337 341 298 076 259
Building50errs.csv trainRMSE testRMSE 1.9757 4.0314 2.7139 6.2222 2.7835 5.4576 2.9771 6.8797 3.1093 4.7846 3.6101 4.5342 1.6541 5.5567 3.0225 7.3988
date 001 243 107 337 341 298 076 259
Building53errs.csv trainRMSE testRMSE 13.0682 26.4726 6.0205 14.1705 6.4943 12.7095 11.4501 26.5387 15.8296 16.2554 6.4616 10.8375 3.5744 10.2883 6.4255 12.1282
date 001 243 107 337 341 298 076 259
V. A NALYSIS The models using the clustering did not perform as well as expected compared to the models using just the historical data of that particular building being analyzed. This may be that a building’s profile is quite unique and surround buildings may not be good features for predictions. It might also be the case that the clustering could be better. Further plots and analysis of the clustered dates were completed. The data to suggests that the days being clustered do follow a logic in that each has a pattern. The plots of cluster k=8 with cluster 1 is displayed. The mean Kw corresponds to the mean vector of the matrix from the first half of the data, as described in the k-means section.
C. GPR with similar building The 32 models used an input matrix of size (7*96 = 672)x(number of buildings in cluster) to predict the output vector of size 96. Each of the test buildings were from a building cluster, so their respective building clusters were used as the input. Thus, a vector of the time period (672) from each building in the test building’s cluster that is not the test building was used and the input matrices’ sizes differed depending on the number of buildings in the cluster. date 001 243 107 337 341 298 076 259
Building27errs.csv trainRMSE testRMSE 2.2500 2.8123 0.6901 7.9039 0.8641 6.9301 1.0063 17.3021 0.7230 6.7764 2.1044 4.9508 0.1623 6.5125 0.0086 6.5019
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 0.2706 1.5223 0.0758 3.2870 0.0972 17.0242 0.0653 24.6378 0.0016 16.3858 0.0005 19.8603 0.0041 20.5427 0.0018 19.8467
date 001 243 107 337 341 298 076 259
Building41errs.csv trainRMSE testRMSE 2.3962 4.9584 3.8441 7.2553 5.2072 9.3673 3.9320 10.9341 4.6400 6.8721 3.6479 7.3121 0.8671 9.5621 3.8320 11.0443
Building50errs.csv trainRMSE testRMSE 16.2173 17.9709 7.3066 19.9443 9.6251 14.6105 13.0713 21.4529 9.3861 19.3867 9.2154 9.7162 0.0250 16.5415 8.7231 13.4898
Errors might also be due to having only 7 days. The author wanted to use more than 7 days, but GPR is a non-parameteric model, which means it is very memory intensive and Matlab gave errors that ‘the array exceeds maximum array size preference’. More data points also slows the for loop from 1:96 that retrains a model per run to be slow and given 32 models to per experiment. VI. C ONCLUSION While the results from the experiments seem to suggest that similar days are not quite that important, it might be due to the clustering of the buildings, so more work needs to be done to understand why similar buildings do not have a high predictive value.
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R EFERENCES Carl Edward Rasmussen and Chris Williams. Gaussian Processes for Machine Learning. Cambridge, Massachusetts: The MIT Press, 2006. Carl Edward Rasmussen and Chris Williams. The GPML Toolbox version 3.5. manual for GPML Matlab Code version 3.6. July 2015.
